Question: Edith wants to make a paperweight at pottery class. She wants the paperweight to have a pyramid-like shape with a base area of $120$ square centimeters, and she wants it to weigh $400$ grams. The density of the clay she is using is $1.6$ grams per cubic centimeter. What should be the height of the paperweight?
Answer: This is a density word problem. To solve it, we can use the following equation, which is the volume definition of density: ${\text{Density}}=\dfrac{{\text{Total quantity}}}{{\text{Volume}}}$ What do we know? The ${\text{density}}$ of the clay is ${1.6}$ grams per cubic centimeter. The paperweight's weight should be ${400}$ grams, which is the ${\text{total quantity}}$. The base area of the pyramid-like paperweight is $120$ square centimeters (we can use this to find the ${\text{volume}}$ ). What do we need to find? The height of the paperweight, which together with the base area gives us the ${\text{volume}}$ of the paperweight. Let's denote the height of the pyramid-like paperweight as $h$. Then, the ${\text{volume}}$ is $\dfrac13 \cdot120h={40h}$ cubic centimeters. Now we can plug ${\text{density}=1.6}$, ${\text{total quantity}=400}$, and ${\text{volume}=40h}$ in the equation. $\begin{aligned} {\text{Density}}&=\dfrac{{\text{Total quantity}}}{{\text{Volume}}} \\\\ {1.6}&=\dfrac{{400}}{{40h}} \\\\ {40h}\cdot{1.6}&=\dfrac{{400}}{\cancel{{40h}}}\cdot\cancel{{40h}} \\\\ 64h&=400 \\\\ h&=\dfrac{400}{64}=6.25 \end{aligned}$ The height of the paperweight should be $6.25$ centimeters.